#501568

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6,8,9)(7,11,12,10)(13,14,19,32,28,36,30,39,16,20,17,22,18,23,34,43)(15,24,41,37,31,44,29,21,25,42,26,40,27,38,33,35);(1,2)(3,4)(6,9)(10,11)(14,20)(21,35)(22,32)(23,36)(24,42)(37,40)(38,44)(39,43);(2,4)(13,15,16,25)(14,21,20,35)(17,29,19,33)(18,31,28,27)(22,38,32,44)(23,40,36,37)(24,43,42,39)(26,34,41,30);(5,7,8,12)(6,10,9,11)(14,20)(21,35)(22,32)(23,36)(24,42)(37,40)(38,44)(39,43)

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408764);